Heisenberg scaler

ABSTRACT

A Heisenberg scaler reduces noise in quantum metrology and includes: a stimulus source that provides physical stimuli; a physical system including quantum sensors that receive a first and second physical stimuli; produces a measured action parameter; receives an perturbation pulse; and produces modal amplitude; an estimation machine that: receives the measured action parameter and produces a zeroth-order value from the measured action parameter; a gradient analyzer that: receives the measured action parameter and produces the measured action parameter and a gradient; the sensor interrogation unit that; receives the modal amplitude; receives the gradient and the measured action parameter produces the perturbation pulse; and produces a first-order value from the modal amplitude, the gradient, and the measured action parameter; a Heisenberg determination machine that: receives the zeroth-order value; receives the first-order value; and produces a physical scalar from the zeroth-order value and the first-order value.

CROSS REFERENCE TO RELATED APPLICATIONS

The application claims priority to U.S. Provisional Patent ApplicationSer. No. 62/758,333 filed Nov. 9, 2018, the disclosure of which isincorporated herein by reference in its entirety.

This invention was made with United States Government support from theNational Institute of Standards and Technology (NIST), an agency of theUnited States Department of Commerce and under Agreement No.W911NF1520067 awarded by the Army Research Lab, Agreement No.W911NF1410599 awarded by the Army Research Office, and Agreement No.W911NF16-1-0082 awarded by the Intelligence Advanced Research ProjectsActivity IARPA). The Government has certain rights in the invention.Licensing inquiries may be directed to the Technology PartnershipsOffice, NIST, Gaithersburg, MD, 20899, voice (301) 301-975-2573; emailtpo@nist.gov; reference NEST Docket Number 19-006US1.

BRIEF DESCRIPTION

Disclosed is a process for determining a physical scalar of an arbitraryresponse fimetion, the process comprising: providing the arbitraryresponse function that comprises a plurality of action parameters θi;subjecting a physical system that comprises a plurality of quantumsensors to a physical stimulus; producing, for an action parameter ofeach quantum sensors, in response to subjecting the quantum sensors tothe physical stimulus, a measured action parameter to provide aplurality of measured action parameters for the physical system;producing a zeroth-order value of the arbitrary response function byevaluating the arbitrary response function at the measured actionparameters; determining the gradient of the arbitrary response functionat the measured action parameters; producing an perturbation pulse;subjecting the .physical system to the perturbation pulse; producing, inresponse to the perturbation pulse, modal amplitude comprising ameasured value of a dot product of the gradient and a vector of actionparameters θi; producing a first-order value of the arbitrary responsefunction by subtracting from modal amplitude the dot product of thegradient and the vector of measured action parameter; and adding thezeroth-order value and the first-order value to determine the physicalscalar of the arbitrary response function.

Disclosed is a Heisenberg scaler for reducing noise in quantummetrology, the Heisenberg scaler comprising: a stimulus source thatprovides a first physical stimulus and a second physical stimulus; aphysical system in communication with the stimulus source and comprisinga plurality of quantum sensors and that: receives the first physicalstimulus and the second physical stimulus from the stimulus source;produces measured action parameter in response to receipt of the firstphysical stimulus; receives an perturbation pulse from a sensorinterrogation unit; produces modal amplitude; an estimation machine incommunication with the physical system and that: receives the measuredaction parameter from the physical system; and produces a zeroth-ordervalue from the measured action parameter; a gradient analyzer incommunication with the physical system and the sensor interrogation unitand that: receives the measured action parameter from the physicalsystem; and produces the measured action parameter and a gradient fromthe measured action parameter; the sensor interrogation unit incommunication with the physical system and the gradient analyzer andthat: receives the modal amplitude from the physical system; receivesthe gradient and the measured action parameter from the gradientanalyzer; produces the perturbation pulse; and produces a first-ordervalue from the modal amplitude, the gradient, and the measured actionparameter; a Heisenberg determination machine in communication with theestimation machine and the sensor interrogation unit and that: receivesthe zeroth-order value from the estimation machine; receives thefirst-order value from the sensor interrogation unit; and produces aphysical scalar from the zeroth-order value and the first-order value.

BRIEF DESCRIPTION OF THE DRAWINGS

The following description should not be considered limiting in any way.With reference to the accompanying drawings, like elements are numberedalike.

FIG. 1 shows a Heisenberg scaler;

FIG. 2 shows a physical system;

FIG. 3 shows a physical system including a quantum sensor network ofspatially separated nodes, wherein at each node, action parameter θi iscoupled to a qubit that accumulates phase proportional to θi;

FIG. 4 shows a physical system including a quantum sensor network ofseparate interferometers, wherein in each interferometer, an armaccumulates an unknown phase θi and the other arm is a reference portwith no phase;

FIG. 5 shows a physical system including a quantum sensor network offield- quadrature displacement sensors, wherein each sensor includes afield mode that experiences a displacement by a real field quadratureθi, as well as a homodyne detector that measures this real quadrature;

FIG. 6 shows a system for dete ling a modal amplitude of aninhomogeneous field of an anaiyte;

FIG. 7 shows a quantum sensor;

FIG. 8 shows a system for determing a modal amplitude of aninhomogeneous field of an analyte;

FIG. 9 shows a graph of difference in energy versus field strength;

FIG. 10 shows a graph of amplitude of a versus time in panel A, andpanel B shows a change from an initial entangled state to a finalentangled state in response to subjecting a quantum sensor to aterminating pulse;

FIG. 11 shows a change from an initial entangled state to a finalentangled state in response to subjecting a quantum sensor to aterminating pulse;

FIG. 12 shows a graph of amplitude of an echo pulse versus time in panelA, and panel B shows a change from an initial entangled state to a finalentangled state in response to subjecting a quantum sensor to an echopulse;

FIG. 13 shows a change from an, initial, entangled state to a finalentangled state in response to subjecting a quantum sensor to an echopulse;

FIG. 14 shows a graph of amplitudes of terminating pulses versus timefor jecting a quantum sensor to a plurality of terminating pulses;

FIG. 15 shows a change from an initial entangled state to a finalentangled state through an intermediate entangled state in response tosubjecting a quantum sensor to a plurality of terminating pulses;

FIG. 16 shows a graph of amplitudes of perturbation pulses versus timefor subjecting a quantum sensor to a plurality of echo pulses;

FIG. 17 a change from an initial entangled state to a final entangledstate through an intermediate entangled state in response to subjectinga quantum sensor to a plurality of echo pulses;

FIG. 18 shows a graph of amplitudes of perturbation pulses versus timefor subjecting a quantum sensor to a plurality of perturbation pulses;

FIG. 19 shows a change from an initial entangled state to a finalentangled state through an intermediate entangled state in response tosubjecting a quantum sensor to a plurality of perturbation pulses; and

FIG. 20 shows a change from an initial entangled state to a finalentangled state through an intermediate entangled state in response tosubjecting a quantum sensor to a plurality of perturbation pulses.

DETAILED DESCRIPTION

A detailed description of one or more embodiments is presented herein byway of exemplification and not limitation.

It has been discovered that a Heisenberg scaler and processes hereinoptimally include quantum entanglement in a network of quantum sensorsto optimally measure a smooth function of fields at the quantum sensors.It is contemplated that in applications for geodesy, geophysics,biology, medicine, and the like, wherein sensors can be separated by aselected distance to measure temperature, a field (e.g., magnetic field,electric field, or a combination thereof), pressure, and the like, theHeisenberg scaler and processes apply when the fields at the sensors aredifferent such as a sensor that measures electric field and anothersensor that measures temperature. The process and Heisenberg scaler caninclude an array of interferometers, and an array of field-quadraturedisplacement sensors, measuring functions of parameters some of whichare measured by sensors, while others are measured by interferometers,while others are measured by field-quadra ure displacement sensors, andthe like.

Fields at individual sensors or phases of individual interferometers orfield-quadrature displacements are measured without entanglement betweensensors or interferometers or field-quadrature displacement sensors to aprecision sufficient for linearization of a function. The resultinglinearized function is measured optimally by distributing a selectedentangled state across the network of sensors or interferometers orfield-quadrature displacement sensors. For qubits, a first step ofentanglement-free measurements can be done in a time proportional toT{circumflex over ( )}(3/5), where T is the total available time, andthe fraction of time spent on this first step vanishes as T becomeslonger and longer. Thus, time is spent on the optimal measurement of thelinearized function. For interferometers and field-quadraturedisplacement sensors, a first step of entanglement-free measurements canbe done with a number of photons proportional tom^({circumflex over ( )}()3/5), where m is the total number of photonsavailable for the measurement, and the faction of photons used in thisfirst step vanishes as m becomes larger and larger. The photons arespent on the optimal measurement of the linearized function.

Advantageously, the process and Heisenberg scaler use entanglement tooptimally measure linear combinations of fields at the N sensors (qubitsensors, interferometers, or field-quadrature displacement sensors) andmeasure arbitrary smooth functions so that, for qubit sensors, a desiredmeasurement is performed at a Sqrt[N] times faster than anentatudement-free methodology, and so that, for interferometers andfield-quadrature displacement sensors, the desired measurement isperformed by a number of photons that is Sqrt[N] times smaller than anentanglement-free methodology. Reducing the required number of photonsis particularly relevant when analyzing a biological or chemical samplethat is sensitive to light, making it desirable to reduce noise with asfew photons as possible. For a fixed time, in the case of sensors or fora fixed number of photons in the case of interferometers andfield-quadrature displacement sensors, a measurement uncertainty isSqrt[N] times smaller than an entanglement-free methodology.Beneficially, the process and Heisenberg scaler measures properties, ofinhomogeneous fields and functions that depend on a measurable quantity.

Heisenberg scaler 300 reduces noise in quantum metrology and determinesphysical scalar 250 of arbitrary response function 210. In anembodiment, with reference to FIG. 1, FIG. 2, FIG. 3, and FIG. 4,Heisenberg scaler 300 includes: stimulus source 228 that provides firstphysical stimulus 216.1 and second physical stimulus 216.2; physicalsystem 200 in communication with stimulus source 228 and including aplurality of quantum sensors 2 and that: receives first physicalstimulus 216.1 and second physical stimulus 216.2 from stimulus source228; produces measured action parameter 218 in response to receipt offirst physical stimulus 216.1; receives perturbation pulse 8 from sensorinterrogation unit 240; produces modal amplitude 238; estimation machine232 in communication with physical system 200 and that: receivesmeasured action parameter 218 from physical system 200; and produceszeroth-order value 220 from measured action parameter 218; gradientanalyzer 236 in communication with physical system 200 and sensorinterrogation unit 240 and that: receives measured action parameter 218from physical system 200; and produces measured action parameter 218 andgradient 252 from measured action parameter 218; sensor interrogationunit 240 in communication with physical system 200 and gradient analyzer236 and that: receives modal amplitude 238 from physical system 200;receives gradient 252 and measured action parameter 218 from gradientanalyzer 236; produces perturbation pulse 8; and produces first-ordervalue 226 from modal amplitude 238, gradient 252, and measured actionparameter 218; and Heisenberg determination machine 234 in communicationwith estimation machine 232 and sensor interrogation unit 240 and that:receives zeroth-order value 220 from estimation machine 232; receiveszeroth-order value 220 from estimation machine 232; receives first-ordervalue 226 from sensor interrogation unit 240; and produces physicalscalar 250 from zeroth-order value 220 and first-order value 226.

Stimulus source 228 can include a pulse source that provideselectromagnetic radiation having a frequency near resonance with thequbit sensor. Exemplary stimulus sources 228 include pulse sources suchas lasers) of optical frequencies for optical qubits, pulse sources ofmicrowave frequencies for microwave qubits, and pulse sources of ratioliequencies for radiofrequency qubits. In the case of interferometers,stimulus source 228 is a source of photons, wherein the photons arecommunicated to the interferometer to determine the phases θi. In thecase of field-quadrature displacement sensors, the stimulus source 228is a source of squeezed light that is sent to each displacement sensorin order to measure the displacements θi.

Stimulus source 228 produces physical stimulus 216 that can include apulse of electromaimetic radiation having a frequency near resonancewith the qubit sensor, Exemplary physical stimuli 216 include opticalpulses (such as laser pulses) for optical qubits, microwave pulses formicrowave qubits, and radiofrequency pulse for radiofrequency qubits. Apower of physical stimulus 216 can be chosen to implement a pi-over-2pulse on the qubit sensor. As used herein, “power” refers to the area ofthe pulse, which includes both the time and the intensity. It iscontemplated that physical system 200 is subjected to first physicalstimulus 216.1 and subsequently to second physical stimulus 216.2. Inthis regard first physical stimulus 216.1 changes the quantum state of aqubit sensor from initial state |0> to final state (|0>+|1>)/Sqrt[2].First physical stimulus 216.1 can include a pi-over-2 pulse ofelectromagnetic radiation. Further, second physical stimulus 216.2 canbe a pi-over-2 pulse on a qubit sensor that maps the phase picked by thequbit on a population difference between |0> and |1>, which can then beprojectively measured by, for example, scattering photons and observingfluorescence. In the case of interferometers, physical stimulus 216 is aphotonic state sent to the interferometer in order to determine thephases θi. The n photons are prepared in state (|n,0>+|0,n>)/sqrt[2],where the first entry denotes the mode that picks up the phase θi, whilethe second entry denotes the reference mode that picks up no phase. Inthe case of field-quadrature displacement sensors, physical stimulus 216is squeezed state of light that is sent to each displacement sensor inorder to measure the displacements Ili. The light is squeezed in thequadrature that is being displaced.

Physical system 200 can include an array or network of qubit sensors, anarray or network of interferometers, an array of network offield-quadrature displacement sensors, or a combination thereof.

Physical system 200 includes a plurality of quantum sensors 2 that caninclude a two-level quantum system such as provided by qubits, athree-level quantum system such as provided by qutrits, a four-levelquantum system, . . . , an m-level quantum system and the like, whereinin is an integer. It is contemplated that energy differences aremeasured between two levels so certain embodiments are described in thecontext of qubits. Exemplary quantum sensors 2 include a nuclear spin,an electronic spin, any two chosen levels Of a neutral atom, an ion, amolecule, a solid-state defect, a superconducting qubit, and the like.In an embodiment, quantum sensors 2 include a neutral atom, an ion, amolecule, a solid-state defect (such as color center in diamond), asuperconducting circuit, and the like, or a combination thereof. Theenergy differences θi between the two levels of each qubit sensor candepend linearly on an observable of interest such as an electric field,a magnetic field, a gravitational field, temperature, strain, and thelike. These observables of interest can be produced by an analyte thatcan include a planet, an organism (e.g. a human), an organ (e.g, a brainor a heart), a tissue (e.g., cardiac tissue), a laser, a molecule (e.g.including macromolecule such as a protein ora nucleic acid), an atom andthe like. In the case of interferometers, the quantum sensor 2 is theinterferometer including a path that goes through the medium of interestand picks up a phase θi and a reference path that doesn't pick up aphase. The medium of interest can include a tissue, a cell, or any othermedium that transmits light. In the case of field-quadrattnedisplacement sensors, quantum sensors 2 can include a bosonic mode thatundergoes a field-quadrature displacement θi and a bornodvue detectorused to measure this field quadrature. The bosonic mode can describemechanical motion where θi can be proportional to a force. The bosonicmode can describe photons where θi can be proportional to a magneticfield via Faraday-rotation after passing through the medium. The bosonicmode can describe low-energy excitations of a large number of two-levelatoms where θi can be proportional to an applied electric or magneticfield.

Physical system 200 produces measured action parameter 218 that caninclude estimates of action parameters θi 212. For qubit sensors,measured action parameter 218 can include estimates of energydifferences between the two levels of the qubit. For interferometers,measured action parameter 218 can include estimates of phases in thearms of the interferometers. For field-quadrature displacement sensors,measured action parameter 218 can include estimates of thedisplacements.

Estimation machine 232 receives measured action parameter 218, andestimation machine 232 can include a classical computer that evaluatesthe desired arbitrary response function 210 f(θ1, . . . , θN) at themeasured action parameter 218. Estimation machine 232 produceszeroth-order value 220 that can include the desired arbitrary responsefunction 210 401, . . . , θN) evaluated at the measured action parameter218.

Gradient analyzer 236 receives measured action parameter 218, andgradient analyzer 236 can include a classical computer that evaluatesthe gradient ∇f of the desired arbitrary response function 210 f at themeasured action parameter 218. Gradient analyzer 236 produces measuredaction parameter 218, as well as the gradient 252 that is the gradientΔf({{tilde over (θ)}_(i)}) of the desired arbitrary response function210 f evaluated at the measured action parameter 218 {tilde over(θ)}_(i).

Sensor interrogation unit 240 receives the measured action parameter 218and gradient 252, and sensor interrogation unit 240 can include a pulsesource 6 which produces perturbation pulse 8 in the form of pulses tocontrol the sensor qubits. Exemplary sensor interrogation units 240include pulse sources (such as lasers) of optical frequencies foroptical qubits, pulse sources of microwave frequencies for microwavequbits, and pulse sources of radiofrequencies for radiofrequency qubits.For interferometers, the pulse source is a source of entangled photonsthat are sent to the interferometers in order to measure the quantity∇f({{tilde over (θ)}_(i)})·{right arrow over (θ)}. For field-quadraturedisplacement sensors, the pulse source is a source of multimodeentangled squeezed light that is sent to the displacement sensors inorder to measure the quantity ∇f({{umlaut over (θ)}_(i)})·{right arrowover (θ)}.

Sensor interrogation unit 240 produces perturbation pulse 8 that caninclude optical pulses (such as laser pulses) for optical qubits,microwave pulses for microwave qubits, and tadiofrequency pulse forradiofrequency qubits. Exemplary perturbation pulse 8 includes pulsesthat entangle qubits into the initial state (|00 . . . 0>+|11 . .1>)/Sqrt[2] via phonon-mediated interactions, electric intataCtiOnS,magnetic interactions, or interactions mediated by flying photonicqubits. Moreover, while this entangled state of qubits is picking upphases proportional to θ1, exemplary perturbation pulse 8 includespulses that are used to control this entangled state of qubits in such away that the picked up phase is proportional to ∇f({{umlaut over(θ)}_(i))·{right arrow over (θ)}, which is the dot product of thegradient 252 and the vector of θi. In the case of interferometers, theperturbation pulse 8 includes the multi-mode entangled photonic state(|n1,0,n2,0, . . . >+|0,n1,0,n2, . . . >)/sqrt[2], where, in thereference to FIG. 4, the modes are listed from top to bottom and wherenj=n*aj/(sum_k a_k), where a_k is the k'th component of ∇f(∇{umlaut over(θ)}_(i))} and where n is the total number of photons we have at ourdisposal. In the case of a combination of qubits and interferometers,the entangled state to be used is an entangled state of photons andqubits of the form (|n1,0,n2,0, . . . >|111 . . . >+|0,n1,0,n2, . .. >|000 . . . >)/sqrt[2].

Sensor interrogation unit 240 receives modal amplitude 238 that caninclude the measured value of ∇f({{umlaut over (θ)}_(i)})·{right arrowover (θ)}, which is the dot product of the gradient 252 and the vectorof θi.

Sensor interrogation unit 240 produces first-order value 226 that caninclude the difference between the modal amplitude 238, i.e. ∇f({{umlautover (θ)}_(i)})·{right arrow over (θ)}, and ∇f({{umlaut over(θ)}_(i)})·({umlaut over (θ)}₁, {umlaut over (θ)}₂, . . . , {umlaut over(θ)}_(N)), which is the dot product of gradient 252 and the vector ofmeasured action parameter 218 {umlaut over (θ)}_(i).

Heisenberg determination machine 234 receives first-order value 226 andzeroth-order value 220, and Heisenberg determination machine 234 caninclude a classical computer that adds the first-order value 226 andzeroth-order value 220 to produce the physical scalar 250.

Heisenberg determination machine 234 produces Physical scalar 250 thatcan include the sum of the first-order value 226 and the zeroth-ordervalue 220.

Physical scalar 250 is determined for arbitrary response function 210 ofquantum sensors 2 in physical system 200. Arbitrary response function210 can include an arbitrary analytic, function f of θi. Exemplaryarbitrary response functions 210 include (1) an arbitrary implicitfunction, which is a solution to some equation that has no closed formsolution that can be easily evaluated, (2) a function obtained, byassuming that θi have some functional form that depends on certaintunable ansatz parameters and the position of the sensor (which could becubit sensor, an interferometer, or a displacement sensor), and thenusing interpolation to express the value of this function at a positionwhere there is no sensor in terms of its values θi at the positions ofthe sensors, (3) a function obtained by assuming that θi have somefunctional form that depends on certain tunable ansatz parameters andthe position of the sensor, and inverting this function to solve for oneof the ansatz parameters in terms of θi, (4) a function obtained byrunning supervised machine learning on training data consisting ofinput-output pairs, (5) a function obtained by using interpolation toinfer the intensity of a laser beam at the position of a trapped dataion lased on the intensities of the laser beam at the positions oftrapped sensor ions, and the like.

Heisenberg scaler 300 can be made in various ways. In an embodiment, aprocess tbr making Heisenberg scaler 300 includes: disposing physicalsystem 200 in communication with stimulus source 228; disposingparameter analysis machine 230 in communication with physical system200; disposing estimation machine 232 in communication with parameteranalysis machine 230; disposing gradient analyzer 236 in communicationwith parameter analysis machine 230 disposing sensor interrogation unit240 in communication with gradient analyzer 236; disposing sensorinterrogation unit 240 in communication with physical system 200;disposing Heisenberg determination machine 234 in communication withestimation machine 232; and disposing Heisenberg determination machine234 in communication with sensor interrogation unit 240.

Heisenberg scaler 300 has numerous advantageous and unexpected benefitsand uses. In an embodiment, a process for determining physical scalar250 of arbitrary response function 210 includes: providing arbitraryresponse function 210 that includes a plurality of action parameters θi212 by explicitly giving the mathematical formula for the function or byspecifying the function implicitly as a solution to sonic set ofequations; subjecting physical system 200 that includes a plurality ofquantum sensors 2 to physical stimulus 216; producing a plurality ofmeasured action parameters 218 for physical system 200 in response tosubjecting the quantum sensors 2 to the physical stimulus 216; producingzeroth-order value 220 of arbitrary response function 210 by evaluatingarbitrary response function 210 at measured action parameters 218;producing the gradient 252 that is the gradient of arbitrary responsefunction 210 evaluated at the measured action parameter 218; producingthe modal amplitude 238 for physical system 200 in response tosubjecting the quantum sensors 2 to the perturbation pulse 8; producingthe first-order value 226 from modal amplitude 238, gradient 252, andmeasured action parameter 218; and combining zeroth-order value 220 andfirst-order value 226 to determine physical scalar 250 of arbitraryresponse function 210.

The process for determining physical scalar 250 of arbitrary responsefimction 210 also can include producing perturbation pulse 8, subjectingphysical system 200 to perturbation pulse 8; and producing modalamplitude 238 in response to perturbation pulse 8.

In an embodiment, with reference to FIG. 6, physical system 200 isdisposed proximate to analyte 4 such that physical system 200 issubjected to inhomogeneous field 50 of analyte 4. Additionally, sensorinterrogation unit 240 includes pulse source 6 that providesperturbation pulse 8 to physical system 200. Physical system 200determines modal amplitude q of inhomogeneous field 50 of analyte 4 inresponse to receipt of perturbation pulse 8.

In an embodiment, as shown in panel. A of FIG. 7, physical system 200includes a plurality of qudit sensrs, e.g., first qudit sensor WA andsecond qudit sensor 10B. it is contemplated that physical system 200 caninclude an arbitrary number N of qudit sensors as shown in panel B ofFIG. 7.

In a particular embodiment, with reference to FIG. 8, analyte 4 can beplanet 62 that includes a plurality of continents 60 (e.g., firstcontinents 60A, second continents 60B), wherein planet 62 hasinhomogeneous field 50, e.g., an inhomogeneous magnetosphere. Physicalsystem 200 is disposed proximate to planet 62 in a presence ofinhomogeneous field 50 such that qudit sensors 10 (e.g., 10A, 10B) aresubjected to inhomogeneous field 50, here the inhomogeneousmagnetosphere. Further, pulse source 6 (not shown) provides perturbationpulse 8 (not shown) to qudit sensors 10 (10A, 10B). As a result, sensorinterrogation unit 240 determines the modal amplitude q of inhomogeneousfield 50 of planet 62.

In physical system 200, first qudit sensor WA includes a plurality ofquantum levels that are subject to entanglement with a plurality ofquantum levels of second qudit sensor JOB to provide an initialentangled state of physical system 200. The initial entangled stateincludes a first quantum level, a second quantum level, and an enentydifference between the second quantum level and the first quantum level.Here, the energy difference between the first quantum level and thesecond quantum level of the initial entangled state is linearlydependent on a strength of inhomogeneous field 50 as shown in FIG. 9.Moreover, the initial entangled state is an initial linear superpositionof the first quantum level and the second quantum level.

Qudit sensor 10 can be a two-level quantum system such as provided byqubits, a three-level quantum system such as provided by qutrits, afour-level quantum system, . . . , an m-level quantum system, and thelike, wherein in is an integer. Accordingly, qudit sensor 10 can includea qubit, qutrit, a quartit, and the like, or a combination of suchqudits. Exemplary qubits include a nuclear spin-1/2, an electronicspin-1/2, or any two chosen levels of a neutral atom, an ion, amolecule, a solid-state defect, a superconducting circuit, and the like.Exemplary qutrits include a spin-1 particle or any three chosen levelsof a neutral atom, an ion, a molecule, a solid-state defect, asuperconducting circuit, and the like. In an embodiment, qudit sensor 10include a neutral atom, an ion, a molecule, a solid-state detect (suchas a nitrogen vacancy color center in diamond), a superconductingcircuit, and the like, or a combination.

In an embodiment, first qudit sensor 10A is a qubit, and second quditsensor 10B is a qubit. In an embodiment, first qudit sensor 10A is aqutrit, and second qudit sensor 10B is a qutrit. In an embodiment, firstqudit sensor 10A is a qubit, and second qudit sensor 10B is a qutrit

It is contemplated that qudit sensors 10 can be included in, physicalsystem 200 as a monolithic device, wherein physical system 200 cantinclude the substrate in which qudit sensors 10 are disposed andarranged. Here, qudit sensors 10 are in mechanical communication viaphonons in the substrate or interact via electric or magneticinteractions. Qudit sensors 10 alternatively can be arranged in anetwork such that the plurality of qudit sensors 10 are subject toquantum entanglement using flying photonic qubits.

It is contemplated that the initial entangled state of physical system200 can be changed to an intermediate entangled state or final entangledstate in response to receipt of perturbation pulse 8 by physical system200. In an embodiment, sensors 10 include first qutrit 10A and secondqutrit 10B, wherein the initial entangled state is the first linearsuperposition

${\frac{1}{\sqrt{2}}\left\lbrack {{00\rangle} + {11\rangle}} \right\rbrack},$

which then evolves under the coupling to the inhomogeneous field to

$\frac{1}{\sqrt{2}}\left\lbrack {{00\rangle} + {e^{{- i}\; \vartheta_{1}}{11\rangle}}} \right\rbrack$

for some phase θ₁ and the final entangled state is final linearsuperposition

$\frac{1}{\sqrt{2}}\left\lbrack {{00\rangle} + {e^{{- i}\; \vartheta_{1}}{21\rangle}}} \right\rbrack$

which then also evolves under the coupling to the inhomogeneous field to

${\frac{1}{\sqrt{2}}\left\lbrack {{00\rangle} + {e^{{- i}\; t_{f}q}{11\rangle}}} \right\rbrack}.$

Here ket |00

refers to both qutrits in state |0

; ket |11

refers to both qutrits in state |0

; ket |21

the first (path in state |2

and the second qutrit in state|1

; tf is the total evolution time and q is the modal amplitude that isbeing measured.

According to an embodiment, qudits 10 include first qubit 10A and secondqubit 10B, wherein the initial entangled state is, an initial linearsuperposition

${\frac{1}{\sqrt{2}}\left\lbrack {{000\rangle} + {111\rangle}} \right\rbrack}.$

The intermediate entangled state includes intermediate linearsuperposition

$\frac{1}{\sqrt{2}}\left\lbrack {{100\rangle} + {e^{{- i}\; \vartheta_{2}}{011\rangle}}} \right\rbrack$

for some phase θ₂ picked up due to coupling to the inhomogeneous field.The final entangled state includes final linear superposition

${\frac{1}{\sqrt{2}}\left\lbrack {{110\rangle} + {e^{{- i}\; t_{f}q}{001\rangle}}} \right\rbrack}.$

Physical system 200 is subjected to inhomogeneous field 50 of analyte 4.Exemplary analytes include a planet, an organism (e.g., a human) anorgan (e.g., a brain, heart, and the like), a tissue (e.g., cardiactissue), a molecule (e.g., including a macromolecule such as a proteinor nucleic acid), an atom, and the like.

Inhomogeneous field 50 of analyte 4 includes modal amplitude q that isdetermined by physical system 200 in response to receipt of perturbationpulse 8 by physical system 200 from pulse source 6. Exemplaryinhomogeneous fields 50 include an electric field, magnetic field,temperature, gravitational field, strain, and the like, or a combinationthereof.

Pulse source 6 provides perturbation pulse 8 to physical system 200.Perturbation pulse 8 can be electromagnetic radiation having a frequencynear resonance with the qudit, such as, for example, optical frequenciesfor optical qubits, microwave frequencies fbr microwave qubits, orradiofrequencies for radiofrequency qubits. A duration of perturbationpulse 8 is short enough so that inhomogeneous field 50 has negliitibleeffect on qudit 10 during perturbation pulse 8. Perturbation pulse 8produced by pulse source 6 is received by an individual qudit sensor(e.v, 10A, 10B, . . . 10N). In response to receipt of perturbation pulse8 by qudit sensor 10 (e.g., 10A, 10B), the first entangled state ischanged to an intermediate entangled state or final entangled state ofphysical system 200.

Exemplary pulse sources 6 include a laser, a microwave source, aradiofrequency source, or a combination thereof.

Physical system 200 can be made in various ways. In an embodiment, aprocess for making physical system 200 includes disposing a first quditsensor 10A at a first position relative to analyte 4 and disposingsecond qudit sensor 10B at a second position relative to analyte 4 andfirst qudit sensor 10A. According to an embodiment, a process for makingphysical system 200 includes providing a substrate; disposing firstqudit sensor 10A in the substrate; and disposing second qudit sensor 10Bin the substrate at a selected position relative to first qudit sensor10A.

Physical system 200 has numerous beneficial uses, including determiningmodal amplitude q of inhomogeneous field 50 of analyte 4. in anembodiment, a process for determining modal amplitude q of inhomogeneousfield 50 of analyte 4 includes: preparing an initial entangled state ofphysical system 200, the initial entangled state including: a firstquantum level; a second quantum level; and an energy difference betweenthe second quantum level and the first quantum level, the energydifference being linearly dependent on a strength of inhomogeneous field50, the initial entangled state being an initial linear superposition ofthe first quantum level and the second quantum level, and physicalsystem 200 including a plurality of qudit sensors 10; subjectingphysical system 200 to inhomogeneous field 50 of analyte 4; subjectingfirst qudit sensor 10A of quantum sensor 10 to a first perturbationpulse; receiving the first perturbation pulse by first qudit sensor 10Ato prepare a first intermediate entangled state of physical system 200,the first intermediate entangled state comprising a first intermediatelinear superposition; changing the initial linear superposition to diefirst intermediate linear superposition in response to receiving thefirst perturbation pulse by physical system 200; and determining a finalentangled state of physical system 200 after applying the firstperturbation pulse to determine modal amplitude q of inhomogeneous field50 of analyte 4.

In the process, preparing an initial entangled state of physical system200 includes preparing a pure unentangled (i.e., product) state of thequdit sensors. It is contemplated that preparing the initial entangledstate of the quantum sensor includes subjecting the qudits to directentangling interaction among the qudits. Here, the interaction could be,for example, electromagnetic interaction such as electric or magneticdipole-dipole interaction or a van der Waals interaction. In an aspect,use of interactions between two of the qudits can prepare an entangledstate shared between them, and then an interaction between a third quditand one of the first two qudits can be used to add the third qudit tothe entangled state, then a fourth qudit, and the like. Preparing theinitial entangled state of the quantum sensor also can includesubjecting the quilts to a mediator comprising a photon, a phonon, or acombination thereof. Here, one can entangle a given qudit with amediator and then send the mediator to the second qudit. Alternatively,one can entangle two qudits with their own mediators and then send themediators towards each other for a joint measurement. One can thenrepeat this process to add additional qudits to the entangled state.

In the process, subjecting physical system 200 to inhomogeneous field 50of analyte 4 includes placing qudit sensors at positions where theknowledge of the inhomogeneous field of interest is needed.

In the process, subjecting first qudit sensor 10A of quantum sensor 10to a first perturbation pulse includes, applying to the qudit sensor ashort near-resonant pulse of electromagnetic radiation that has just theright pulse area (equal to π) to transfer the qudit state from |1

to |2

for the case of a terminating pulse or to swap qudit levels |1

and |2

for the case of an echo pulse.

In the process, receiving the first perturbation pulse by first quditsensor 10A to prepare a first intermediate entangled state of physicalsystem 200 includes transferring the qudit state from |1

to |2

for the case of a terminating pulse or swapping qudit levels |1

and |2

for the case of an echo pulse.

In the process, changing the initial linear superposition to the firstintermediate linear superposition in response to receiving the firstperturbation pulse by physical system 200 includes transferring thequdit state from |1

to |2

for the case of a terminating pulse or swapping qudit levels |1

and |2

for the case of an echo pulse.

In the process, determining a final entangled state of physical system200 after applying the first perturbation pulse and after waiting forthe final time includes undoing all termination and echo pulses and thenprojectively measuring each qudit in the

$\frac{1}{\sqrt{2}}\left\lbrack {{0\rangle} \pm {1\rangle}} \right\rbrack$

basis and then multiplying the answers (plus or minus one) to computethe parity P, whose quantum mechanical expectation value depends onmodal amplitude q as

P

cos(qtf). One can then extract q by repeating the experiment many times.Here, modal amplitude q includes a linear combination of a plurality ofmode components α_(i) of a mode A and a plurality of energy componentsθi

${q = {\sum\limits_{i}^{N}{\alpha_{i}\theta_{i}}}},$

wherein N is a total number of qudits 10, and i is an integer from 1 toN.

With reference to FIG. 10 and FIG. 12, the first pertubation pulse canbe a termination pulse as shown in FIG. 10 or an echo pulse as shown inFIG. 12. Further, the process can include subjecting second qudit sensor10B of quantum sensor 10 to a second perturbation pulse; receiving thesecond perturbation pulse by second qudit sensor 10B to prepare thesecond intermediate entangled state of physical system 200, the secondintermediate entangled state including a second intermediate linearsuperposition; changing the first intermediate linear superposition tothe second intermediate linear superposition in response to receivingthe second perturbation puke by physical system 200; and determining thefinal entangled state of the quantum sensor after applying the secondperturbation pulse to determine modal amplitude q of inhomogeneous field50 of analyte 4. In this regard, FIG. 14, FIG. 16, and FIG. 18 show thesecond perturbation pulse occurring after the first perturbation pulse.

In an embodiment, subjecting first qudit sensor 10A to the firstperturbation pulse occurs at a time based on a smallest mode component(i.e., the least αi) of modal amplitude q. It is contemplated that modecomponents αi can be scaled to the greatest mode components αi andreordered according to magnitude, so that the minimum mode component isαl, and the maximum mode component is αN. In this manner, the firstperturbation pulse occurs at first time t1 based on first mode componentα1, and second perturbation pulse occurs at second time t2 based onsecond mode component α2. Swapping the labels of |0

and |1

allows one to deal with negative values of αi. If αi are complex, oneWould repeat the entire procedure twice-once for the real parts, andonce for the imaginaiy parts.

The perturbation pulse can be a terminating pulse or echo pulse. Withreference to FIG. 10, which shows a terminating pulse in panel A, aninitial entangled state of physical system 200 is prepared at time t0=0,and qudit sensor 10 of physical system 200 is subjected to a terminatingpulse at first time t1=α_(i)t_(r), which is a time multiple of i-th modecomponent α_(i). As a result, as shown in panel B of FIG. 10, theinitial entangled state is changed to a final entangled state inresponse to receipt of the terminating pulse. Here, the initialentangled state can include, e.g., initial linear superposition of firstquantum level |0

and second quantum level |1

, wherein the initial entangled state is 1/√2[|0

+|1÷] having energy difference E=θ between first quantum level |0

and second quantum level |1

. The terminating pulse at time t1 transitions second quantum level |1

to third quantum level |2

, which is degenerate with first quantum level |0

, to produce the final entangled state as 1/√2[|0

+|2

] having energy difference E=0. For convenience, the state of the quditsensor is written as a pure state, but it is actually part of anentangled state with other qudit sensors. Moreover, it also contains aphase due to coupling to inhomogeneous field 50. After time t1, thefinal entangled state of physical system 200 is determined and used todetermine modal amplitude q of inhomogeneous field 50 produced byanalyte 4.

In an embodiment, with reference to FIG. 11, physical system 200includes a first qudit sensor 10A (having quantum levels |0

and |1

with energy difference E=θz,34 and second qudit sensor 10B (havingquantum levels |0

and |1

with energy difference E=θ2), wherein physical system 200 has initialentangled state 1/√2[|00

+|11

] having energy difference E=θ1+θ2 between first quantum level |00

and second quantum level |11

. A first terminating pulse is subjected to first qudit sensor 10A attime t1=α_(i)t_(r) to produce final entangled state 1/√2[|00

+e^(−it) ¹ ^((θ1+θ2))|21

] having energy difference E=θ2, where the phase was acquired during thetime t1 via coupling to inhomogeneous field 50. This entangled statethen evolves for time tf−t1 to produce state 1/√2[|00

+e^(itfq)|21

]. It is convenient to undo the terminating pulse at the end to givestate 1/√2[|00

+e^(−itfq)|11

], which is then measured.

In an embodiment, perturbation pulse includes an echo pulse. Withreference for FIG. 12, which shows an echo pulse in panel A, an initialentangled state of physical system 200 is prepared at time t0, and quditsensor 10 of physical system 200 is subjected to an echo pulse at firsttime t1=a_(i)t_(r)+c, which is after a time multiple of i-th modecomponent

by a selected time amount c=tf (1−α_(i))/2. As a result, as shown inpanel B of FIG. 12, the initial entangled state is changed to a finalentangled state in response to receipt of the echo pulse. Here, theinitial entangled state can include, e.g., initial linear superpositionof first quantum level |0

and second quantum level |1

, wherein the initial entangled state is 1/√2[|0

+|1

] having energy difference E=θ between first quantum level |0

and second quantum level |1

. The echo pulse at time t1 exchanges second quantum level |1

with first quantum level |0

to produce the final entangled state as 1/√2[|1

+|0

] having energy difference E=−θ. For convenience, the state of the quditsensor is written as a pure state, it is actually part of an entangledstate with other qudit sensors and it also contains a phase picked updue to coupling to inhomogeneous field 50. After time t1, the finalentangled state of physical system 200 is determined and used todetermine modal amplitude q. of physical system 200.

In an embodiment, with reference to FIG. 13, physical system 200includes a first qudit sensor 10A (having quantum levels |0

and |1

with energy difference E=θ1

and second qudit sensor 10B (having quantum levels |0

and |1

with energy difference E=θ2), wherein physical system 200 has initialentangled state 1/√2[|00

+|11

] having energy difference E=θ1+θ2 between first quantum level |00

and second quantum |11

. A first echo pulse is subjected to first qudit sensor 10A at timet1=α_(i)t_(r)+c in which levels of first qudit sensor 10A are exchanged(i.e., |0

⇄|1

) to produce final entangled state 1/√2[|10

+e^(−it) ¹ ^((θ1+θ2))|01

] having energy difference E=−θ1+θ2, where the phase was acquired duringthe time t1 via coupling to the inhomogeneous field. This entangledstate then evolves for time tf−t1 to produce state 1/√2[|00

+e^(−itfq)|01

]. It is convenient to undo the echo pulse at the end to give state1/√2[|00

+e^(−itfq)|11

], which is then measured.

In an embodiment, with reference to FIG. 14, physical system 200including a plurality of qudits 10 (e.g 10A, 10B, and the like) issubjected to preparation of an initial entangled state at tune t0=0,which is subjected to a plurality of perturbation pulses, e.g.,terminating pulses. At time t1, first qudit 10A is subjected to a firstterminating pulse to prepare first intermediate entangled state. At timet2, second qudit 10B is subjected to a second terminating pulse toprepare a final entangled state that is subjected to determination forobtaining modal amplitude q. Here, as shown in FIG. 15, the initialentangled state of physical system 200 can be 1/√2[|000

+|111

] such that first terminating pulse at time t1 produces intermediateentangled state 1/√2[|000

+e^(−it) ¹ ^((θ1+θ2+θ3))|211

], which is subjected to second terminating pulse at time t2 to producefinal entangled state 1/√2[|000

+e^(−it) ¹ ^((θ1+θ2+θ3)−i(t) ² ^(−t) ¹ ^()(θ2+θ3))|221

]. This entangled state then evolves for time tf-t2 to produce state1/√2[|000

+e^(itfq)|221

]. It is convenient to undo the terminating pulses at the end to givestate 1/√2[|000

+e^(−itfq)|111

], which is then measured.

In an embodiment, with reference to FIG. 16, physical system 200including a plurality of qudits 10 (e.g 10A, 10B, and the like) issubjected to preparation of an initial entangled state at time t0, whichis subjected to a plurality of perturbation pulses e.g., echo pulses. Attime t1, first qudit 10A is subjected to a first echo pulse to preparefirst intermediate entangled state. At time t2, second qudit 10B issubjected to a second echo pulse to prepare a final entangled state thatis subjected to determination for obtaining modal amplitude q. Here, asshown in FIG. 17, the initial entangled state of physical system 200 canbe 1/√2[|000

+|111

] such that first echo pulse at time t1 produces intermediate entangledstate 1/√2[|100

+e^(−it) ¹ ^((θ1+θ2+θ3))|011

], which is subjected to second echo pulse at time t2 to produce finalentangled state 1/√2[|110

+e^(−it) ¹ ^((θ1+θ2+θ3)−(″θ1+θ2+θ3)) ∥001

]. This entangled state then evolves for time tf−t2 to produce state1/√2[|111

]. It is convenient to undo the echo pulses at the end to give state1/√2[|000

+e^(−itfq)|111

], which is then measured.

In an embodiment, with reference to FIG. 18, physical system 200including a plurality of qudits 10 (e.g 10A, 10B, and the like) issubjected to preparation of an initial entangled state at time t0, whichis subjected to a plurality of perturbation pulses e.g., a combinationof echo pulses and terminating pulses. At time t1, first qudit 10A issubjected to an echo pulse to prepare first intermediate entangledstate. At time t2, second qudit 10B is subjected to a terminating pulseto prepare a final entangled state that is subjected to determinationfor obtaining modal amplitude q. is contemplated that, as shown in FIG.19, the initial entangled state of physical system 200 can be 1/√2[|000

+|111

] such that the echo pulse at time t1 produces intermediate entangledstate 1/√2[|100

+e^(−it) ¹ ^((θ1+θ2+θ3))|011

], which is subjected to terminating pulse at time t2 to produce finalentangled state 1/√2[|100

+e^(−it) ¹ ^((θ1+θ2′θ3)−i(t) ² ^(−t) ¹ )(−θ1+θ2+θ3)|021

]. This entangled state then evolves for time tf−t2 to produce state1/√2[|100

+e^(−itfq)|021

]. It is convenient to undo the perturbation pulses at the end to givestate 1/√2[|000

+e^(−itfq)|111

], which is then measured.

In a plurality of perturbation pulses, a sequence of the perturbationpulses can be a selected sequence of terminating pulses in the echopukes in a selected number of the terminating pulses and echo pulses. Inan embodiment, with reference to FIG. 20, initial entangled state ofphysical system 200 can be 1/√2[|000

+|111

] such that the terminating pulse at time t1 produces intermediateentangled state 1/√2[|000

+e^(−it) ¹ ^((θ1+θ2+θ3))|211

], which is subjected to an echo pulse at time t2 to produce finalentangled state 1/√2[|010

+e^(−it) ¹ ^((θ1+θ2+θ3)−i(t) ₂−t₁)(θ2+θ3)|201

]. This entangled state then evolves for time tf−t2 to produce state1/√2[|010

+e^(−itfq)|201

]. It is convenient to undo the perturbation pulses at the end to givestate 1/√2[|000

+e^(−itfq)|111

], which is then measured.

The articles and processes herein are illustrated further by thefollowing Example, which is non-limiting

EXAMPLE

Heisenberg-scaling measurement process for analytic functions withquantum sensor networks.

For d input parameters, entanglement can yield a factor O(d) improvementin mean-squared error when estimating an analytic function of theseparameters. The process is optimal for qubit sensors and for photonspassing through interferometers. The applies to continuous variablemeasurements, such as one quadrature of a field operator. Applicationsinclude calibration of laser operations in trapped ion quantumcomputing.

Entanglement is a resource for quantum technology. In metrology,entangled probes provide more accurate measurements than unentangledprobes. In addition to using entangled probes to enhance the measurementof a single parameter, using entanglement to estimate many parameters atonce, or a function of those parameters, has applications for nanosealenuclear magnetic resonance imaging.

A lower bound on the variance of an estimator of a linear combination ofd parameters coupled to d qubits is provided. We generalize measurementof an arbitrary real-valued, analytic function of d parameters and showthat entanglement reduces variance of such an estimate by a factor ofO(d). Described is a process that provides optimal varianceasymptotically in the limit of long measurement time. In addition, whenthe parameters are coupled to d interferometers or to a combination ofinterferometers and qubits, an analogous Heisenberg-scaling process toimprove measurement noise is provided. The process couples parameters tocontinuous variables detected by homodyne measurements.

The process applies to field interpolation. Suppose sensors are placedat d spatially separated locations, but we wish to know the field at apoint with no sensor. We may pick a reasonable ansatz for the field withno more than d parameters, use our d measurements to fix the degrees offreedom of that ansatz, and compute the field at our desired point.Because the field of interest is a function of the field at d otherlocations, the process offers reduced noise over performing the sameprocedure without using entanglement.

Here, bold face font indicated vectors, hats (as in Ĥ) indicateoperators, and variables with a tilde (such as {tilde over (f)}) areestimators of the corresponding quantity with no tilde (such as f). Thenotation E_(Y)[X] means the expected value of X over all possible Y. Ifwe merely write E[X], then we average over all parameters required todefine X (e.g. if Y depended on Z, then E_(z)[E_(y)[X]]). We define thevariance, Var_(y)[X] similarly.

We consider a system with d sensor nodes, where node i consists of asingle qubit coupled to a real parameter θ_(i) (see FIG, 3), and supposethat the state evolves under the Hamiltonian

$\begin{matrix}{\mspace{79mu} {{\hat{H} = {{{\hat{H}}_{c}(t)} + {\frac{1}{2}\theta_{t}\text{?}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (1)\end{matrix}$

where ô_(i) ^(x,y,z) are the Pauli operators acting on qubit i andH_(c)(t) is a time-dependent control Hamiltonian that we choose, whichmay include coupling to ancilla qubits. Here, and throughout the paper,repeated indices indicate summation. We want to measure an arbitraryreal-valued, analytic function f(θ) of d unknown parameters θ=(θ₁ . . .θ_(d)) for time t_(total). Determine how well the quantity f(θ) an beestimated and find a process for doing so. To specify a process, wechoose an input state, a control Hamiltonian H_(c)(t), and a finalmeasurement.

For a general estimator, we use the mean squared error (MSE) M of ourestimate {tilde over (f)} from the true vale f(θ) as a figure of merit.Explicitly,

$\begin{matrix}{M = {{E\left\lbrack \left( {\overset{\sim}{f} - {f(\theta)}} \right)^{2} \right\rbrack} = {{{Var}\overset{\sim}{\; f}} + {\left( {{E\left\lbrack \overset{\sim}{f} \right\rbrack} - {f(\theta)}} \right)^{2}.}}}} & (2)\end{matrix}$

Thus, the MSE accounts for both the variance and the bias of theestimator {tilde over (f)}. By proving lower-hounds for M and thenshowing that these bounds are saturable, we will be demonstratingprocess which are optimal in this combination of bias and variance.

With regard to a lower hound on error, we now identify the minimumpossible error of an estimator of f (θ) which measures for time For anyestimator {tilde over (f)}, biased or otherwise, which uses samples froma probabilistic process (such as physical experiments) to estimate thevalue f(θ), the MSE is bounded by

$\begin{matrix}{{{\left\lbrack \left( {\overset{\sim}{f} - {f(\theta)}} \right)^{2} \right\rbrack} \geq \frac{1}{F} \geq \frac{1}{F_{Q}}},} & (3)\end{matrix}$

where F is the. Fisher information for the parameter f and F_(Q) is thequantum Fisher information evaluated over our input state, with F_(Q)≥Falways. Bounds on the error of an estimator in terms of the Fisherinformation are known as Cramér-Rao bounds. The Fisher informationmeasures the sensitivity of the sampled probability distribution tochanges in the parameters θ. While F tells us something about aparticular experimental setup, F_(Q) is maximized over all possibleexperiments that could be performed on a state.

In order to evaluate the Fisher information for our function of interestf, we will use a method for linear functions. We start by evaluating thegenerator ĝ=∂Ĥ.∂f. By first writing the chain rule, we find that

$\begin{matrix}{\hat{g} = {\frac{\partial\hat{H}}{\partial f} = {{\frac{\partial\hat{H}}{\partial\theta_{i}}\frac{\partial\theta_{i}}{\partial f}} = {\frac{1}{2}{\hat{\sigma}}_{z}^{i}{\frac{\partial\theta_{i}}{\partial f}.}}}}} & (4)\end{matrix}$

Note that F_(Q) can be upper-bounded by the seminorm of this generator,F_(Q)≤t²|ĝ|_(s) ². (The seminorrn of an operator is the differencebetween its maximum and minimum eigenvlues.) However, to evaluate theseminorm we will need to evaluate the partial derivative in Eq. (4). Todo so we must specify a full basis of functions so that the partialderivative can be defined, which requires specifying which variables areheld constant during differentiation. We suppose that a set of functionsf₁, f₂, f₃ . . . f_(N) are created, with the f of interest equal to f₁,defining an invertible coordinate transformation on a region R^(N)around the point θ. The seminorin is then:

$\begin{matrix}{{\hat{g}}_{s} = {{\sum\limits_{i = 1}^{N}{\frac{\partial\theta_{i}}{\partial f}}} = {\sum\limits_{i = 1}^{N}{{J_{i\; 1}^{- 1}}.}}}} & (5)\end{matrix}$

Here, J_(ij) ⁻¹ is an element of the Jacobian matrix of the inversetransformation to that defined by the f functions. Depending on whichfunctions are chosen, the value of |ĝ|_(s), can vary for linearfunctions. We therefore wish to find the smallest possible |ĝ|_(s),which will provide the tightest possible bound on F_(Q). To do so, wenote that J⁻¹ and J must obey an inverse relationship, meaning that thefollowing chain of inequalities holds,

$\begin{matrix}{1 = {{J_{1i}J_{i\; 1}^{- 1}} \leq {{J_{1i}}{J_{i\; 1}^{- 1}}} \leq {\max\limits_{j}{{J_{1j}}{\sum\limits_{i = 1}^{N}{{J_{i\; 1}^{- 1}}.}}}}}} & (6)\end{matrix}$

By using the definition of the Jacobian, we can rewrite this as a lowerbound on the value of |ĝ|_(s) in terms of partial derivatives of f:

$\begin{matrix}{{\hat{g}}_{s} = {{\sum\limits_{i = 1}^{N}{J_{i\; 1}^{- 1}}} \geq {\left( {\max\limits_{j}{\frac{\partial f}{\partial\theta_{j}}}} \right)^{- 1}.}}} & (7)\end{matrix}$

If we label the θ_(i) yields the maximum first derivative as θ₁, andthen choose f_(i)=θ_(i) for i>1, the lower bound in Eq. (7) is met since∂θ_(i)/∂f₁ must be evaluated holding the other f_(i) constant. Invokingthe resulting bound on the quantum Fisher information, we find that thequantum Cramér-Rao bound becomes

$\begin{matrix}{M = {{\left\lbrack \left( {\overset{\sim}{f} - {f(\theta)}} \right)^{2} \right\rbrack} \geq \frac{1}{F_{Q}} \geq {\max\limits_{j}{\frac{{\frac{\partial f}{\partial\theta_{j}}}^{2}}{t^{2\;}}.}}}} & (8)\end{matrix}$

Although the quantum Cramér-Rao bound derived in Eq. (8) cannot alwaysbe saturated, it can when the generators ∂H/∂θ_(i) commute, as in Eq.(1). We will show later that the inequality in Eq. (8) can be saturatedat asymptotic time t_(total).

From this point forward, to simplify later calculation, we define

${f_{i\;}(\theta)} = {\frac{\partial{f(\theta)}}{\partial\theta_{i}}.}$

This definition also generalizes to multiple partial derivatives (i.e.

$\left. {f_{ij} = {\frac{\partial}{\partial\theta_{j}}\frac{\partial f}{\partial\theta_{i}}}} \right).$

Before moving on to the optimal process, we will consider a processwhich does not use entanglement and does not saturate Eq. (8) as auseful contrast to an entangled strategy. Suppose we estimate eachparameter individually, without bias. Then the MSE

[(f({tilde over (θ)})−f(θ)²] can be written as

M_(unentangled)=f_(i)(θ)²Var{tilde over (θ)}_(i)  (9)

Here we assume the measurement of each single parameter can be made intime t with Var

=1/t{circumflex over ( )}2, the Heisenberg limit for single particlesand therefore the best possible measurement for a non-entangled process.Estimation processes allow one to reach a vnriance proportional to 1/t²without entanglement; an experimental realization of single phaseestimation without entanglement was performed. While in realisticsettings a Heisenberg-limited measurement on one particle may bechallenging and include some constant overhead above 1/t², thisassumption allows us to identify the improvement possible by usingentanglement. Our entanglement-free figure of merit is

$\begin{matrix}{{M_{unentangled} = \frac{{{\nabla{f(\theta)}}}^{2}}{t_{{total}^{2}}}},} & (10)\end{matrix}$

where the ∥·∥ in Eq. (10) denotes the Euclidean norm. More generally, weuse ∥v∥p to denote the p-norm of vector v. Since Eq. (10) only saturatesEq. (8) in trivial cases where ∇f/(θ) is zero in all but one component,the the entangled process described is not optimal.

In a two-step process, we now present a process which asymptoticallysaturates Eq. (8). Our process consists of two steps. First, we make anunbiased estimate {tilde over (θ)}θ for time t₁. Second, given ourestimates {tilde over (θ)}, we make an unbiased measurement {tilde over(q)} the quantity q=∇f(θ)·(θ−{tilde over (θ)}) using a linearcombination process, which takes time t₂. Our final estimate is {tildeover (f)}=f({tilde over (θ)})+{tilde over (q)}.

It can be shown that our process is optimal (in terms of scaling withthe total time t₁t₂) provided that the individual estimations of theparameters satisfy

[

−θ_(i))⁴]=O(t₁ ⁻⁴) and that hand t₁ and t₂ are chosen properly. Tosimplify our computations, we will make the more concrete assumptionthat our initial estimates {tilde over (θ)} are each normallydistributed as

(0,Var{tilde over (θ)}_(i)). The figure of merit for this process is

$\begin{matrix}\begin{matrix}{M = {\left\lbrack \left( {{f\left( \overset{\sim}{\theta} \right)} + \overset{\sim}{q} - {f(\theta)}} \right)^{2} \right.}} \\{= {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\frac{{2{f_{ij}(\theta)}} + {{f_{ij}(\theta)}{f_{jj}(\theta)}}}{4}{Var}\; {\overset{\sim}{\theta}}_{i}{Var}\; {\overset{\sim}{\theta}}_{j}}}}\end{matrix} & {(11)(12)}\end{matrix}$

In Eq. (12), fhe first term is the error resulting from the second phaseof the process, estimating the linear combination. The second term is aresidual error remaining from the first phase of the process after it iscorrected by the linear combination measurement, For our particularHamiltonian

we know that the minimum variance of an unbiased estimator of somelinear combination α·θ given timer is

$\begin{matrix}{{{Var}\; {\alpha \cdot \theta}} \geq \frac{\max\limits_{i}\alpha_{i}^{2}}{t^{2}}} & (13)\end{matrix}$

which can be achieved with the entangled GHZ state

${\psi_{spin}\rangle} = {\frac{1}{\sqrt{2}}{\left( {{0\rangle}^{\otimes d} + {1\rangle}^{\otimes d}} \right).}}$

We can apply this linear combination process to the second phase of ourprocess by setting α=∇f({umlaut over (θ)}). For the individualestimators of the first phase, we use the filet that an individualestimation can be made in time t with variance 1/t². Using theseresults, we simplify Eq. (12);

$\begin{matrix}{M = {{\left\lbrack \frac{\max_{i}{f_{i}\left( \overset{\sim}{\theta} \right)}^{2}}{t_{2}^{2}} \right\rbrack} + \frac{\frac{{2{f_{ij}\left( \overset{\sim}{\theta} \right)}} + {{f_{ii}\left( \overset{\sim}{\theta} \right)}{f_{jj}\left( \overset{\sim}{\theta} \right)}}}{4}}{t_{i}^{4}}}} & (14) \\{M = {\frac{\left\lbrack {\max\limits_{1}{f_{1}\left( \overset{\sim}{\theta} \right)}^{2}} \right\rbrack}{t_{2}^{2}} + \frac{g_{1}\left( \overset{\sim}{\theta} \right)}{t_{1}^{4}}}} & (15)\end{matrix}$

where we have absorbed the second derivatives into g₁(θ), which does notdepend on time.

Without loss of generality, we designate f₁({tilde over (θ)}) as thelargest f_(i) ({tilde over (θ)}) We then expand

[f_(i)({tilde over (θ)})] as

$\begin{matrix}{{f_{1}(\theta)}^{2} + \frac{{f_{1}(\theta)}{f_{1{ii}}(\theta)}}{t_{1}^{2}} + \frac{{f_{1i}(\theta)}^{2}}{t_{1}^{2}} + {{\left( \left( {\overset{\sim}{\theta} - \theta} \right)^{3} \right)}.}} & (16)\end{matrix}$

We may substitute Eq. (16) into Eq. (15) to obtain

$\begin{matrix}{{M = {\frac{g_{2}(\theta)}{t_{2}^{2}} + \frac{g_{3}(\theta)}{t_{1}^{2}t_{2}^{2}} + \frac{g_{1}(\theta)}{t_{1}^{4}} + {\left( \left( {\overset{\sim}{\theta} - \theta} \right)^{3} \right)}}},} & (17)\end{matrix}$

where g₂(θ)=f₁(θ)² and g₃(θ) have been introduced to absorb moretime-independent factors.

With regard to optimal time allocation, to complete the process, wespecify how the total time t_(total) is to be allocated between t₁ handt₂. We want to choose the t₁, t₂, under the constraint thatt₁+t₂=t_(total), which minimize the MSE

$\begin{matrix}{M = {\frac{g_{2}(\theta)}{t_{2}^{2}} + \frac{g_{3}(\theta)}{t_{1}^{2}t_{2}^{2}} + {\frac{g_{1}(\theta)}{t_{1}^{4}}.}}} & (18)\end{matrix}$

The g₁, g₂, g₃ functions are only dependent on θ and not t₁, so we mayset the derivative of M with respect to t₁ equal to 0 and obtain

$\begin{matrix}{{\frac{2{g_{2}(\theta)}}{t_{2}^{3}} + \frac{2{g_{3}(\theta)}}{t_{2}^{3}t_{1}^{2}}} = {\frac{2{g_{3}(\theta)}}{t_{2}^{3}t_{1}^{3}} + {\frac{4{g_{1}(\theta)}}{t_{1}^{3}}.}}} & (19)\end{matrix}$

Let r=t₁/t₂. Then we may rearrange to obtain

$\begin{matrix}{{{g_{2}(\theta)}t_{1}^{2}} = {\frac{g_{3}(\theta)}{r} + \frac{2{g_{1}(\theta)}}{r^{3}} - {{g_{3}(\theta)}.}}} & (20)\end{matrix}$

Since t₁»1, then r«1, so the r⁻³ term dominates the: RHS. Thus,

${{{g_{2}(\theta)}t_{1}^{2}} \approx \frac{2{g_{3}(\theta)}}{r^{3}}},$

which implies

$\begin{matrix}{t_{1} \approx {\left( \frac{2{g_{1}(\theta)}}{g_{2}(\theta)} \right)^{1/5}t_{2}^{3/5}} \approx {\left( \frac{2{g_{1}(\theta)}}{g_{2}(\theta)} \right)^{1/5}t_{total}^{3/5}}} & (21)\end{matrix}$

Therefore, the best possible allocation satisfies

t₁=g(θ)t_(total,) ^(3/5)  (22)

where g is a function which depends only on f and θ. In particular, t₁=

(t_(total) ^(3/5)), so the fraction of time spent on t₁ vanishes ast_(total)→∞. Almost all of the time is spent on t₂, the linearcombination step of the two-step process. It can readily be shown thatEq. (17) is a symptotically dominated by the first term when this timeallocation is chosen, which (since t₂→t_(total)) is equal to theright-hand-side of the bound in Eq, (8). In other words, thisdistribution of time asymptotically achieves the optimal MSE.

The two-step process exhibits Heisenberg scaling as defined fbrdistributed. sensing. Comparing Eq. (10.) to Eq. (8) shows animprovement of O(d), maximized when all components of f(θ) areapproximately equal. Intuitively, the advantage is maximal when allparameters contribute, but minimal (i.e. no advantage) when only oneparameter affects the function value. Similar behavior was noted in thelinear combination case.

Note that when actually implementing the process, the optimal t₁ isunknown since the function g that determines it depends on the tineparameters θ. However, we optionally do not use the optimalt₁ tosaturate the bound in Eq. (8). If t₁ n is a function ct^(p) _(total) ofthe total time where ½<p<1 and some constant c, then the process willsaturate Eq. (12). Suppose that t₁=ct^(p) _(total) for some ½><p <1 andsonic constant c. Since p<1, we see that

${\lim\limits_{t_{total}\rightarrow\infty}\frac{t_{2}}{t_{total}}} = 1.$

Therefore, we may substitute our n into the MSE formula in Eq. (17) andsimplify:

$\begin{matrix}{{\lim\limits_{t_{total}\rightarrow\infty}M} = {{\lim\limits_{t_{total}\rightarrow\infty}\frac{g_{2}(\theta)}{t^{2}}} + \frac{g_{3}(\theta)}{c^{2}t^{2 + {2p}}} + {\frac{g_{1}(\theta)}{c^{4}t^{4p}}.}}} & (23)\end{matrix}$

Since p=>½, the t² ^(total) term is dominant. Thus, as we definedg₂:=f₁(θ)²=max_(i) f_(r)(θ)² under the assumption that f₁(θ)² wasmaximal, our asymptotic error is

$\begin{matrix}{{M = \frac{\max_{i}{f_{i}(\theta)}^{2}}{t^{2}}},} & (24)\end{matrix}$

which saturates the bound of Eq. (8). Although selecting a non-optimaltime allocation does result in is higher MSE, the additional error isO(t⁻⁴), which is insignificant asymptotically. The two-step process willtherefore be asymptotically optimal for a wide range of timeallocations.

With regard to function measurement in certain physical settings,consider a different physical setting for function estirnation, Ratherthan d qubits which accumulate phase for some tune t, we instead pass uphotons through d Mach-Zelmcler interferometers and accumulate somefixed phase θ_(i) encoded into each interferometer (see FIG. 4). Forsingle parameters, the use of entangled states to reduce noise in thissetting has been explored with multiparameter cases. In this setting,the relevant limitation is the total number of photons used in themeasurement, rather than time. This constraint is particularly relevantwhen analyzing a biological or chemical sample which is sensitive tolight, making it desirable to reduce noise with as few photons aspossible. Similar biologically motivated situations are contemplated.

For photons, a two-step process with similar structure to the processfor qubits yields reduced noise compared to any estimate of fderivedentirely from local measurements, Suppose we allot N₁ photons for thefirst step (individual measurement) and N₂ photons for the second step(linear combination), for a total of N_(total)=N₁+N₂ photons. We againbegin from the general result of Eq. (12). However, the use of photonswhich can be apportioned between modes introduces new structure to theproblem. We need to partition the N₁ photons into N₁=n₁+. . . n_(d),putting n_(i) photons into the i-th interferometer, as some parametersmay affect our final result more than others. Thus, in the second termof Eq. (12), we replace Var{tilde over (θ)}_(i) with

$\frac{1}{n_{1}^{2}}$

instead of

$\frac{1}{t_{1}^{2}}.$

The optimal variance when measuring the linear combination α·θ using Ntotal photons is unknown. However, the optimal variance is conjecturedto be

$\begin{matrix}{{{Var}\mspace{11mu} } \geq \frac{{\alpha }_{1}^{2}}{N^{2}}} & (25)\end{matrix}$

Furthermore, achieving the bound in Eq. (25) involves a proportionallyweighted GHZ state:

${{\psi_{photon}\rangle} = {\frac{1}{\sqrt{2}}\left( {{{n_{1},0,n_{2},{0\mspace{14mu} \ldots}}\mspace{11mu}\rangle} + {{0,n_{1},0,n_{2},\ldots}\mspace{11mu}\rangle}} \right)}},$

where

$n_{i} = {N_{total}\frac{\alpha_{i}}{\sum\alpha_{j}}}$

and where, in reference to FIG. 4, the modes are listed from top tobottom. Note that this will only work for a proportional to somerational vector as photons are discrete. Since Eq. (25) is saturable, wemay simplify the first term of Eq. (12) to obtain

$\begin{matrix}{M = {\frac{\left\lbrack {{\nabla{f(\theta)}}}_{1}^{2} \right\rbrack}{N_{2}^{2}} + \frac{{2{f_{ij}(\theta)}^{2}} + \frac{{f_{ii}(\theta)}{f_{jj}(\theta)}}{4}}{n_{i}^{2}n_{j}^{2}}}} & (26)\end{matrix}$

For fixed f and θ, the

$\frac{1}{n_{i}n_{j}}$

terms in Eq. (26) are minimized for the same ratio of n₁:n₂: . . . :n_(d) regardless of the value of the total number of photons used, N₁.Each term is proportional to N₁ ⁻⁴ multiplied by some function of f, θ,and d. Therefore, the structure of Eq. (26) becomes identical to thestructure of Eq. (17), with N₁ and N₂ replacing t₁ and t₂. As a result,the optimal allocation of photons between N₁ and N₂ will yield N₁=

(N_(total) ^(3/5)) and N₂O(N_(total)), meaning that the N₂ ⁻² term inEq. (26) is dominant asymptotically. Therefore, for photons, we mayasymptotically achieve

$\begin{matrix}{M = {\frac{{{\nabla{f(\theta)}}}_{1}^{2}}{N_{2}^{2}} + {\left( \frac{1}{N_{total}^{12/5}} \right)}}} & (27)\end{matrix}$

This strategy is optimal if the linear combination estimation strategyis optimal. Our optimality result remains true for spins evolving underEq. (1) and for photons that our process is conjectured to be optimal.

Eq. (27) also exhibits Heisenberg scaling. Suppose we were to measureeach parameter individually and then calculate the function. Whenmeasuring the parameters individually, we obtain the same error formulaas Eq. (9), except now we set

${\overset{\sim}{\theta}}_{i} = \frac{1}{n_{i}^{2}}$

Var to get

$\begin{matrix}{M_{unentangled} = {\frac{{f_{i}(\theta)}^{2}}{n_{i}^{2}}.}} & (28)\end{matrix}$

The optimal distribution requires an m proportional to the weightf_(i)(θ)^(2/3), yielding an entanglement-free error of

$\begin{matrix}{M_{unentangled} = {\frac{{{\nabla{f(\theta)}}}_{2/3}^{2}}{N^{2}}.}} & (29)\end{matrix}$

As with gaits, by comparing Eq. (27) with Eq. (29) in the case where allof the f_(i)(θ) are approximately equal, we find that the photonictwo-step process yields a O(d) improvement in error over measuring eachparameter individually. This improvement when all quantities are equallyimportant can be seen for the special case off being a linearcombination. As in the cubit case, the improvement in error is lessenedwhen ∇f(θ) is not approximately equal in all components.

This method can be extended still more generally. Rather than caseswhere the signal is imprinted on photons by a phase shift, we canconsider entanglement-enhanced distributed sensing of continuousvariables by using homodyne measurements. Besides measuring parametersin different physical settings, we may also measure functions ofvariables coupled to spins, phase-shifts of photons, continuousvariables, and any combination of these. In such a. hybrid scenario, wecan still make use of the two-step process. The first step, obtaininginitial estimates for the individual parameters, proceeds equivalently,since the measurements of the spins and of the photons can be viewed asoccurring in parallel. For the linear combination case, we can assumethat the optimal spM and photon input states can be entangled asfollows:

${\psi_{{spin}\text{-}{photon}}\rangle} = {\frac{1}{\sqrt{2}}\left( {{{{n_{1},0,n_{2},{0\mspace{14mu} \ldots}}\mspace{14mu}\rangle} \otimes {\rangle}} + {{{1,1,1,\ldots}\mspace{14mu}\rangle} \otimes {{0,{{n\_}1},0,{{n\_}2},\ldots}\mspace{14mu}\rangle} \otimes {{0,0,0,\ldots}\mspace{14mu}\rangle}}} \right)}$

Here,

${n_{i} = {N_{total}\frac{\alpha_{i}}{\sum\alpha_{j}}}},$

where the sum runs over only the j corresponding to photonic modes,denotes the number of photons Which pass through the arms of the i-thinterferometer. The state in Eq. (30) is designed in such a way that thetwo branches of the overall wavefunction accumulate relative to eachother a phase equal to the total linear combination we are interestedin. In order to extract this final phase, the state can be unitarilymapped onto a qubit, which contains all of the accumulated phase and is;then measured,

The linear combination process can accumulate phase proportional to timefbr the quints and phase proportional to the number of photons forinterferometers. For instance, if θ₁ coupled to a qubit (and thereforehas units of frequency) and θ₂ is coupled to an interferometer (and istherefore unitless), then the two branches of our state accumulate arelative phase θ₁f+θ₂n. Therefore, one may have to adjust t or n inorder to get the desired linear combination.

With regard to application, the process estimates an analytic functionof the inputs in a variety of potential applications. When sensors areprocessed into a single signal, the process provides enhancedsensitivity using entanglement. There is no requirement that differentθ₁ have the same physical origin. For instance, a θ₁ representing anelectric field and θ₂ measuring a magnetic field could be used tomeasure the Poynting vector.

One potential application of function measurements is the interpolationof non-linear functions. Suppose that an ansatz with d tunableparameters is made for the strength of the field in a region. Withreadings from≤d different points, one could determine the parameters ofthe ansatz and therefore determine the value of the field at otherpoints. Estimations of these ansatz parameters, which are functions ofthe measured fields, may potentially be improved using entangled statesdepending on the figure of merit. Note that this procedure, can becarried out even if it is difficult to invert the ausatz, in terms ofthe d measurements. Suppose that θ=f(c,x) and that c=f¹(θ, x) exists,but has no closed-form solution which can be easily evaluated. First, wemake measurements {tilde over (θ)}. To create an initial estimate of thevalues c, we use a numerical root-finder to find estimates We can nowimplement the second step of our process by finding the firstderivatives ∂c_(i)/∂θ_(j) using the matrix identity

${\frac{\partial\theta}{\partial c} \cdot \frac{\partial c}{\partial\theta}} = {I.}$

Since f is known, ∂θ/∂c can be inverted to yield the ∂c/∂θ needed toestimate {tilde over (q)}=∂c/∂θ|_(θ={tilde over (θ)})·(θ−{tilde over(θ)}). Our final estimate is {tilde over (c)}+{tilde over (q)}, whichwas obtained without having to compute f⁻¹ in general.

Interpolation in this manner can proceed by two different schemes.Measure the ansatz parameters themselves, which provides computation ofthe field at all other points or skip the final computation step bywriting the field at a point of interest as a function of all the pointsthat can be measured. This final function can be directly measured usingan entangled process, which will be more accurate. The first approachhas the advantage that knowing the ansatz parameters provides estimationof all points in the space in question.

One particular interpolation of interest arises in ion trap quantumcomputing, In trapped ion chains, qubits are manipulated using Gaussianlaser beams, and two primary sources of error are intensity and beampointing fluctuations. Our process offers better ways to characterizethis noise. In order to detect the field error at a cubit's positionwithout disturbing the qubit, we can perform interpolation by measuringthe field's effect on other ions, possibly of a different atomicspecies, positioned nearby. Given the ansate of the Gaussian beamprofile, we are able to calculate the field at the qubit of interest andperhaps correct the error. As entanglement of ions is already afunctionality for trapped ion quantum computers, our process isapplicable in that domain.

The Heisenberg-scaling measurement process includes quantum sensornetworks for measuring any multivariate, real-valued, analytic function,and this process is consistent with the Heisenberg limit when measuringfunctions with comparably-sized gradients in each component. Recentadvances in the distribution of entanglement, for instance, insatellites distributing entangled photons more than 1000 km, strengthenthe viability of this scheme over large distances in the near-term.Potential sensing platforms include trapped ions and nitrogen vacancydefects in diamond, which can also be entangled and are proven platformsfor magnetometry and thermometry. Optimality of the two-step processwhen constrained by the number of photons is contemplated and extendsresults into quantum networking to explore how entanglement can bereliably distributed for metrological purposes.

Field interpolation can be made, and the process can provide measurementof any analytic function.

For derivation of Eq. 12, let Δ={tilde over (θ)}−θ which satisfiesE[Δ]=0. Furthermore, let T_(k) be k! times the k-th term of the Taylorexpansion of f (so T₁=f_(i)(θ)Δ_(i), T₂=f_(ij)(θ)Δ_(i)Δ_(j),T₃=f_(ijk)(θ) Δ_(i)Δ_(j)Δ_(k), etc.). Thus, the Taylor expansion of f({tilde over (θ)}) would be

$\begin{matrix}{{f\left( \overset{\sim}{\theta} \right)} = {{f(\theta)} + T_{1} + \frac{T_{2}}{2} + \frac{T_{3}}{6} + \ldots}} & ({A1})\end{matrix}$

We compute our figure of merit:

$\begin{matrix}{\mspace{20mu} {M = {\left\lbrack \left( {{f\left( \overset{\sim}{\theta} \right)} + \overset{\sim}{q} - {f(\theta)}} \right)^{2} \right\rbrack}}} & ({A2}) \\{\mspace{20mu} {M = {{\left\lbrack \left( {{f\left( \overset{\sim}{\theta} \right)} - {f(\theta)}} \right)^{2} \right\rbrack} + {\left\lbrack \overset{\sim}{q^{2}} \right\rbrack} + {2{\left\lbrack {{f\left( \overset{\sim}{\theta} \right)}q} \right\rbrack}} - {2{f(\theta)}{\left\lbrack \overset{\sim}{q} \right\rbrack}}}}} & ({A3}) \\{{{term}\mspace{14mu} 1\mspace{14mu} {term}\mspace{14mu} 2\mspace{14mu} {term}\mspace{14mu} 3} = {\underset{\underset{{term}\mspace{14mu} 1}{}}{{\left\lbrack T_{1}^{2} \right\rbrack} + {\left\lbrack {T_{1}T_{2}} \right\rbrack} + {\frac{1}{3}{\left\lbrack {T_{1}T_{3}} \right\rbrack}} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right\rbrack}} + {\left( \Delta^{5} \right)}} + \underset{\underset{{term}\mspace{14mu} 2}{}}{{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\left\lbrack q^{2} \right\rbrack}} + {2\; \underset{\underset{{term}\mspace{14mu} 3}{}}{{{f(\theta)}{\lbrack q\rbrack}} + {\left\lbrack {T_{1}q} \right\rbrack} + {\frac{1}{2}{\left\lbrack {T_{2}q} \right\rbrack}} + {\frac{1}{6}{\left\lbrack {T_{3}q} \right\rbrack}} + {\left( \Delta^{5} \right)}}} - {2{f(\theta)}{\lbrack q\rbrack}}}} & ({A4}) \\{= {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\left\lbrack \left( {q + T_{1}} \right)^{2} \right\rbrack} + {\left\lbrack {\left( {q + T_{1}} \right)T_{2}} \right\rbrack} + {\frac{1}{3}{\left\lbrack {\left( {q + T_{1}} \right)T_{3}} \right\rbrack}} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right\rbrack}} + {{\left( \Delta^{5} \right)}.}}} & ({A5})\end{matrix}$

We may simplify

$\begin{matrix}{{q + T_{1}} = {\Delta_{i}\left( {{f_{i}(\theta)} - {f_{i}\left( \overset{\sim}{\theta} \right)}} \right)}} & ({A6}) \\{\mspace{56mu} \begin{matrix}{\; {= {- {\Delta_{i}\left( {{{f_{ij}(\theta)}\Delta_{j}} + {\left( \Delta^{2} \right)}} \right)}}}} \\{= {{- T_{2}} + {\left( \Delta^{3} \right)}}}\end{matrix}} & ({A7})\end{matrix}$

So Eq (5) evaluates to

$\begin{matrix}{M = {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\left\lbrack T_{2}^{2} \right\rbrack} - {\left\lbrack T_{2}^{2} \right\rbrack} - {\frac{1}{3}{\left\lbrack {T_{2}T_{3}} \right\rbrack}} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right\rbrack}} + {\left( \Delta^{5} \right)}}} \\{= {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right\rbrack}} + {\left( \Delta^{5} \right.}}}\end{matrix}$ $\begin{matrix}{M = {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right.}}}} \\{= {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\frac{1}{4}{\left\lbrack \left( {{f_{ij}(\theta)}\Delta_{i}\Delta_{j}} \right)^{2} \right.}}}} \\{= {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\frac{1}{4}{\begin{bmatrix}{{4{\sum\limits_{i < j}{{f_{ij}(\theta)}^{2}\Delta_{i}^{2}\Delta_{j}^{2}}}} + {2{\sum\limits_{i < j}{{f_{ii}(\theta)}{f_{jj}(\theta)}\Delta_{i}^{2}\Delta_{j}^{2}}}} +} \\{\sum\limits_{i}{{f_{ii}(\theta)}^{2}\Delta_{i}^{4}}}\end{bmatrix}}}}}\end{matrix}$

since E[T₂T₃] is O(Δ⁵). Now, this simplifies further as

(A 11)(A 12)(A 13)

since all terms with some Δ_(i) to a single power will factor out asE[Δ_(i)]=0. We will assume that Δ_(i)˜

(0, t⁻²) is normally distributed. This is optional when the distributionof errors satisfies

[(Δ_(i) ⁴]≤t₁ ⁻⁴), a condition that is satisfied by phase estimationprocedures. However, assuming, normality allows the calculation toproceed easily, as we will be able to simplify E[Δ_(i) ⁴]3 Var

. Thus, we arrive at

$\begin{matrix}{M = {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\frac{1}{4}\left( {{4{\sum\limits_{i < j}{{f_{ij}(\theta)}^{2}{Var}\; {\overset{\sim}{\theta}}_{i}{Var}\; {\overset{\sim}{\theta}}_{j}}}} + {2{\sum\limits_{i < j}{{f_{ii}(\theta)}{f_{jj}(\theta)}{Var}\; {\overset{\sim}{\theta}}_{i}{Var}\; {\overset{\sim}{\theta}}_{j}}}} + {\sum\limits_{i}{3f_{ii}\; (\theta)^{2}{Var}\; {\overset{\sim}{\theta}}_{i}^{2}}}} \right)}}} & ({A14}) \\{\mspace{20mu} {= {{\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\sum\limits_{i,j}{\frac{{2{f_{ij}(\theta)}} + {f_{ii}\; (\theta){f_{jj}(\theta)}}}{4}{Var}\; {\overset{\sim}{\theta}}_{i}{Var}\; {{\overset{\sim}{\theta}}_{j}.}}}}}} & ({A15})\end{matrix}$

Here, we present simplification of the labeled terms from Eq. A3-A5).Term 2 is simplified by using the definition ofVar_({tilde over (q)}){tilde over (q)}. One, needs to be careful asthere are tsvo layers of expected values—one for the values of {tildeover (θ)}and one for the estimator {tilde over (q)}:

$\begin{matrix}{{\left\lbrack \overset{\sim}{q^{2}} \right\rbrack} = {_{\overset{\sim}{\theta}}\left\lbrack {_{q}\left\lbrack q^{2} \right\rbrack} \right\rbrack}} & ({A16}) \\{\mspace{59mu} {= {_{\overset{\sim}{\theta}}\left\lbrack {{{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} + {_{\overset{\sim}{q}}\left\lbrack \overset{\sim}{q} \right\rbrack}^{2}} \right\rbrack}}} & ({A17}) \\{\mspace{59mu} {= {_{\overset{\sim}{\theta}}\left\lbrack {{{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} + q^{2}} \right\rbrack}}} & ({A18}) \\{\mspace{59mu} {= {{_{\overset{\sim}{\theta}}\left\lbrack {{Var}_{\overset{\sim}{q}}\overset{\sim}{q}} \right\rbrack} + {\left\lbrack q^{2} \right\rbrack}}}} & ({A19})\end{matrix}$

Terms 1 and 3 are simplified by expanding the Taylor series for f({tildeover (θ)}) up to Δ⁴ terms; note that q=O(Δ), so we only need to expandthe Taylor series up to O(Δ³) terms:

$\begin{matrix}{\underset{\underset{{term}\mspace{14mu} 1}{}}{\left\lbrack \left( {{f\left( \overset{\sim}{\theta} \right)} - {f(\theta)}} \right)^{2} \right\rbrack} = {{\left\lbrack {f\left( \overset{\sim}{\theta} \right)}^{2} \right\rbrack} - {2{f(\theta)}{\left\lbrack {f\left( \overset{\sim}{\theta} \right)} \right\rbrack}} + {f(\theta)}^{2}}} & ({A20}) \\{= {{f(\theta)}^{2} + {\left\lbrack T_{1}^{2} \right\rbrack} + {{f(\theta)}{\left\lbrack T_{2} \right\rbrack}} + {\left\lbrack {T_{1}T_{2}} \right\rbrack} + {\frac{1}{3}{f(\theta)}{\left\lbrack T_{3} \right\rbrack}} + {\frac{1}{12}(\theta){\left\lbrack T_{4} \right\rbrack}} + {\frac{1}{3}{\left\lbrack {T_{1}T_{3}} \right\rbrack}} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right\rbrack}} + {\left( \Delta^{5} \right)}}} & ({A21}) \\{{{{- 2}{f(\theta)}\left( {{f(\theta)} + {\frac{1}{2}{\left\lbrack T_{2} \right\rbrack}} + {\frac{1}{6}{\left\lbrack T_{3} \right\rbrack}} + {\frac{1}{24}{\left\lbrack T_{4} \right\rbrack}} + {\left( \Delta^{5} \right)}} \right)} + {f(\theta)}^{2}} = {{\left\lbrack T_{1}^{2} \right\rbrack} + {\left\lbrack {T_{1}T_{2}} \right\rbrack} + {\frac{1}{3}{\left\lbrack {T_{1}T_{3}} \right\rbrack}} + {\frac{1}{4}{\left\lbrack T_{2}^{2} \right\rbrack}} + {{\left( \Delta^{5} \right)}.}}} & ({A22}) \\{\mspace{20mu} {\underset{\underset{{term}\mspace{14mu} 1}{}}{\left\lbrack {{f\left( \overset{\sim}{\theta} \right)}\overset{\sim}{q}} \right\rbrack} = {{_{\overset{\sim}{\theta}}\left\lbrack _{\overset{\sim}{q}} \right\rbrack}{f\left( \overset{\sim}{\theta} \right)}\overset{\sim}{q}}}} & ({A23}) \\{\mspace{20mu} {= {_{\overset{\sim}{\theta}}\left\lbrack {{f\left( \overset{\sim}{\theta} \right)}q} \right.}}} & ({A24}) \\{\mspace{20mu} {= {\left\lbrack {\left( {{f(\theta)} + T_{1} + \frac{T_{2}}{2} + \frac{T_{3}}{6} + {\left( \Delta^{4} \right)}} \right)q} \right\rbrack}}} & ({A25}) \\{\mspace{20mu} {= {{{f(\theta)}{\lbrack q\rbrack}} + {\left\lbrack {T_{1}q} \right\rbrack} + \frac{\left\lbrack {T_{2}q} \right\rbrack}{2} + \frac{\left\lbrack {T_{3}q} \right\rbrack}{6} + {\left( \Delta^{5} \right)}}}} & ({A26})\end{matrix}$

While one or more embodiments have been shown and described,modifications and substitutions may be made thereto without departingfrom the spirit and scope of the invention. Accordingly, it is to beunderstood that the present invention has been described by way ofillustrations and not limitation. Embodiments herein can be usedindependently or can be combined.

All ranges disclosed herein are inclusive of the endpoints, and theendpoints are independently combinable with each other. The ranges arecontinuous and thus contain every value and subset thereof in the range.Unless otherwise stated or contextually inapplicable, all percentages,when expressing a quantity, are weight percentages. The suffix “s” asused herein is intended to include both the singular and the plural ofthe term that it modifies, thereby including at least one of that term(e.g., the colorant(s) includes at least one colorants). “Optional” or“optionally” means that the subsequently described event or circumstancecan or cannot occur, and that the description includes instances wherethe event occurs and instances where it does not. As used herein,“combination” is inclusive of blends, mixtures, alloys, reactionproducts, and the like.

As used herein, “a combination thereof” refers to a combinationcomprising at least one of the named constituents, components,compounds, or elements, optionally together with one or more of the sameclass of constituents, components, compounds, or elements.

All references ate incorporated herein by reference.

The use of the terms “a” and “an” and “the” and similar referents in thecontext of describing the invention (especially in the context of thefollowing claims) are to be construed to cover both the singular and theplural, unless otherwise indicated herein or clearly contradicted bycontext. “Or” means “and/or.” It should further be noted that the terms“first,” “second,” “primary,” “secondary,” and the like herein do notdenote any order, quantity, or importance, but rather are used todistinguish one element from another. The modifier “about” used inconnection with a quantity is inclusive of the stated value and has themeaning dictated by the context (e.g., it includes the degree of errorassociated with measurement of the particular quantity), The conjunction“or” is used to link objects of a list or alternatives and is notdisjunctive; rather the elements can be used separately or can becombined together under appropriate circumstances.

What is claimed is:
 1. A process for determining a physical scalar of anarbitrary response function, the process comprising: providing thearbitrary responsefunction that comprises a plurality of actionparameters θi; subjecting a physical system that comprises a pla ralityof quantum sensors to a physical stimulus; producing, for an actionparameter of each quantum sensors, in response to subjecting the quantumsensors to the physical stimulus, a measured action parameter to providea plurality of measured action parameters for the physical system;producing a zeroth-order value of the arbitrary response function byevaluating the arbitrary response function at the measured actionparameters; determining the gradient of the arbitrary response functionat the measured action parameters; producing an perturbation pulse;subjecting the physical system to the perturbation pulse; producing, inresponse to the perturbation pulse, modal amplitude comprising ameasured value of a dot product of the gradient and a vector of actionparameters θi; producing a first-order value of the arbitrary responsefunction by subtracting from modal amplitude the dot product of thegradient and the vector of measured action parameter; and adding thezeroth-order value and the first-order value to determine the physicalscalar of the arbitrary response function.
 2. A Heisenberg scaler forreducing noise in quantttna metrology, the Heisenberg scaler comprising:a stimulus source that provides a first physical stimulus and a secondphysical stimulus; a physical system in communication with the stimulussource and comprising a plurality of quantum sensors and that: receivesthe first physical stimulus and the second physical stimulus from thestimulus source; produces measured action parameter in response toreceipt of the first physical stimulus; receives an perturbation pulsefrom a sensor interrogation unit; produces modal amplitude; anestimation machine in cOMITRIllication with the physical system andthat: receives the measured action parameter from the physical system;and produces a zeroth-order value from the measured action parameter; agradient analyzer in communication with the physical system and thesensor interrogation unit and that: receives the measured actionparameter from the physical system; and produces the measured actionparameter and a gradient from the measured action parameter; the sensorinterrogation unit in commt ication with the physical system and thegradient analyzer and that: receives the modal amplitude from thephysical system; receives the gradient and the measured action parameterfrom the gradient analyzer; produces the perturbation pulse; andproduces a first-order value from the modal amplitude, the gradient, andthe measured action parameter; and a Heisenberg determination machine incommunication With the estnmation machine and the sensor interrogationunit and that: receives the zeroth-order value from the estimationmachine receives the first-order value from the sensor interrogationunit; and produces a physical scalar from the zeroth-order value and thefirst-order value.